peno@enea.UUCP (Pekka Nousiainen) (11/25/85)
< line eater bait >
In the November issue of Science '85 Yu.I.Manin writes (p.91) that
Euler's formula
6/PI^2 = (1-1/2^2)*(1-1/3^2)*(1-1/5^2)*...
(2,3,5,... = prime numbers, ^ = exponentiation) "gives the volume of a
simple solid in a space having one ordinary dimension and one arithmetic
dimension". It sounds like he's talking about the space
R x Spec(Z) (R = reals, Z = integers)
where Spec(Z) is the algebro-geometric spectrum of integers. Does
anybody know more about this?
Background for non-mathematicians:
Manin writes about the "odd" dimensions one can add to the ordinary
three dimensions of space. Physicists are using such dimensions to
explain the behavior of elementary particles.
The most familiar (and not "odd") addition of dimension is the complex
numbers: a + bi where i^2 = -1. Similarly one can define numbers a + be
where the new unit e (epsilon) satisfies e^2 = 0. This space, usually
denoted R[e], is already in common use in physics. If you think about
how ordinary numbers behave - 0.1^2 = 0.01, 0.01^2 = 0.0001 - it makes
sense to think of e as being an "infinitely small" number, so small that
its square actually is 0. The resulting space R[e] is sort of real numbers
infinitesimally "thickened" into another dimension of space.
What Manin proposes above is still more abstract (and not yet used in
physics). The points in R x Spec(Z) would be pairs (a, p) where a is
an ordinary number and p is a prime number. There'd also be special
points (a, 0) that sort of glue the whole thing together.
P.S. I have NOTHING against Polar Bears but could we have a bit more
math in this group? The recent posting on the independence of CH via
topos theory is one of the few articles that actually belongs here.
Anybody for net.math.expert?gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (11/26/85)
> 6/PI^2 = (1-1/2^2)*(1-1/3^2)*(1-1/5^2)*... > > (2,3,5,... = prime numbers, ^ = exponentiation) "gives the volume of a > simple solid in a space having one ordinary dimension and one arithmetic > dimension". This formula doesn't even talk about volume, except to the extent that every number can be thought of as the volume of SOMEthing. > It sounds like he's talking about the space > > R x Spec(Z) (R = reals, Z = integers) > > where Spec(Z) is the algebro-geometric spectrum of integers. Does > anybody know more about this? > ... > What Manin proposes above is still more abstract (and not yet used in > physics). The points in R x Spec(Z) would be pairs (a, p) where a is > an ordinary number and p is a prime number. There'd also be special > points (a, 0) that sort of glue the whole thing together. Tell us more about Spec(Z), whose elements are apparently just the positive prime integers. What is this "algebro-geometric spectrum"? I have to object to calling every factor algebra a "dimension".
bill@milford.UUCP (bill) (11/27/85)
> < line eater bait > > > In the November issue of Science '85 Yu.I.Manin writes (p.91) that > Euler's formula > > 6/PI^2 = (1-1/2^2)*(1-1/3^2)*(1-1/5^2)*... > > (2,3,5,... = prime numbers, ^ = exponentiation) "gives the volume of a > simple solid in a space having one ordinary dimension and one arithmetic > dimension". It sounds like he's talking about the space > > R x Spec(Z) (R = reals, Z = integers) > > where Spec(Z) is the algebro-geometric spectrum of integers. Does > anybody know more about this? > In the "Ring of Arithmetic Functions" multiplication is defined as (f * g)(n) = (sum over the divisors of n) of (f(n/d) g(d)) The function e(n) = {1 if n=1; 0 otherwise} turns out to be the identity. The Moebius function 1 if n = 1; mu(n) = { 0 if n has a square dividing it (-1)^q where q is the number of distinct prime divisors This Moebius function then turns out to be the inverse of the constant 1. Further the (infinite sum over n) of (mu(n)/n*n) = 1/zeta(2) = 6/pi*pi. I'd never think of using convolution multiplication for dimensions, but... More of interest to me would be questions like What is the inverse of arithmetic functions like the Euler phi function? I'm not positive if this is what Manin meant (is he on the net? :-))
wdh@faron.UUCP (Dale Hall) (11/28/85)
In article <64@brl-tgr.ARPA> gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) writes: > >Tell us more about Spec(Z), whose elements are apparently just the >positive prime integers. What is this "algebro-geometric spectrum"? > >I have to object to calling every factor algebra a "dimension". Last point first: this isn't an episode of "here's something new! Let's call it a dimension." The notion of dimension for commutative rings (as the maximum length of a chain of prime ideals) is due to Krull (German mathematician, early 20th century [?]), and is relevant to what one would ordinarily think of as the dimension of a topological space. The construction of Spec(R), for a commutative ring R, proceeds as follows: Spec(R): the set of all prime ideals P in R, with a basis of open sets given by the set-theoretic complements (Spec(R) \ V(x)), for x in R; here, the (closed) set V(x) is: V(x) = {Q in Spec(R)| x in Q}. For R = Z (integers), Spec(Z) = {pZ | p :prime}, where 0 is prime, but 1 isn't. The topology restricted to non-zero primes is discrete, but the prime 0 is dense in Spec(Z) - it's called a generic point. The dimension of the prime spectrum Spec(Z) is 1. If k is an algebraically closed field (take your favorite one), and R = k[X], the ring of polynomials in one indeterminate, then Spec(R) is called the affine line over k. Its regular points correspond to maximal ideals in k[X], which are in 1-1 correspondence with the elements of k, and there is again a generic point, given by the zero ideal in k[X]. (this case is similar to that of Spec(Z) - in fact, k[X] is itself similar to Z, as many of you will undoubtedly suspect). The dimension of Spec(k[X]) is also 1. This generalizes to the ring R = k[X ,X ,...,X ], the ring of polynomials in n 1 2 n indeterminates over k; the prime spectrum Spec(R) is then affine n-space over k, and, except for the generic point (guess what?), Spec(R) looks just like the space of n-tuples of elements of k. Then, it can start to get a little technical. (you like sheaves? we got sheaves. you like cohomology? you came to the right place, we got cohomology. how about some nice schemes? a few Chern classes, the Todd genus, and you got a very attractive Riemann-Roch theorem ...) Enjoy. References M. Nagata, Local Rings - Wiley H. Matsumura, Commutative Algebra - Benjamin I.R.Shafarevitch, Basic Algebraic Geometry - Springer Verlag R. Hartshorne, Algebraic Geometry - Springer Verlag
peno@enea.UUCP (Pekka Nousiainen) (11/28/85)
>This formula doesn't even talk about volume, except to the extent >that every number can be thought of as the volume of SOMEthing. You're right. All I can offer is "proof by authority": Manin is one of the best living mathematicians. >Tell us more about Spec(Z), whose elements are apparently just the >positive prime integers. What is this "algebro-geometric spectrum"? >I have to object to calling every factor algebra a "dimension". Sorry about using fancy words. Your objection though isn't valid. A factor algebra, or just any commutative ring A, corresponds to a geometrical object with properties such as dimension and connectivity. Here's an explanation via an [ inaccurate ] example: --begin--technical-- Consider a geometrical object S such as a sphere or the 3-dimensional space. To measure things in S one introduces coordinate functions, that is, continuous functions f: S -> Reals. Such functions can be added and multiplied (just add or multiply the values of the functions) so they form a ring A. One can recover the set S (but not its geometry) from A: a point x in S corresponds to the set of functions f such that f(x) = 0. These sets are exactly the "prime ideals" of A. In a general ring A, the prime ideals play the same role as prime numbers do in integers. The points of Spec(A) are the prime ideals of A. Returning to the example, we have S = Spec(A) as abstract sets. To get further, assume S is the 3-dim space of points (x,y,z). Instead of all continuous functions, take only polynomial functions, such as f(x,y,z) = 3 + x^2*y + z^6. These form another ring B. Now Spec(B) is much more interesting. As in the case of A, prime ideals of B have "zero sets" in S. Any maximal chain of primes in B corresponds to some chain of zero sets looking like ( "-" means inclusion) (point) - (curve) - (surface) - (whole S). You can read off the correct dimension dim(E) = 3 from the structure of B alone (length of longest prime chain). To come to the point, what is the dimension of the integers Z ? The prime ideals are the set consisting of 0, and for each prime number p the set of multiples of p. A longest chain is any 0 - p. The integers are one-dimensional. In math, Z is just another a "curve" (granted, a weird curve). --end--technical-- The above is hardly enough to convince, but the ideas are at the very center of pure mathematics. As for Manin's claim, I suspect Spec(Z) is much too simple (there are others). -- Net address: ...!mcvax!enea!peno.
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (11/30/85)
Thanks for the explanation. This kind of discussion is MUCH more interesting to me than arithmetic puzzles.